Topological Materials: From Weak to Strong Correlations

Today, there is a growing class of magnetic materials where it is believed that the interactions are bond-dependent in a way first imagined by Alexei Kitaev thereby opening a way for realizing topological phases. Bond-dependent interactions are strongly frustrating for the system and hinders conventional ordering. However, in these Kitaev materials other interactions are also often present, among them the well known Heisenberg coupling and also off-diagonal Gamma terms giving rise to an exceptionally rich phase diagram. Even for the simplest models of Kitaev materials it is extremely difficult to arrive at a precise understanding of this complex phase-diagram. Hence, in order to obtain accurate results it is often useful to restrict the analysis to low-dimensions and here we mainly discuss chains and two- (and more) leg ladders. Using numerical techniques, it is possible for such models to determine the phase-diagram with very high precision, including the effects of an applied magnetic field. An astonishing abundance of phases arises from the combination of frustration and applied field. In this talk I will focus on some of these phases that appear disordered, without any conventional local magnetic ordering, but where a hidden string-order can be identified.

In conventional magnetic materials, interactions between the spins lead to a phase transition from a high-temperature disordered state to a magnetically ordered state as the temperature is lowered. The transition is typically accompanied by singularities in the thermodynamic observables at the transition point, and spontaneous symmetry breaking and a reduction of the spin entropy to zero as the system enters a unique ground state. However, the spin entropy can also be released without any symmetry breaking, down to zero temperature, by forming a collective quantum spin state with long-range quantum entanglement. This exotic state of matter is called a quantum spin liquid. A goal of condensed matter physics is to discover new quantum phases formed by the ensemble of interacting spins and charges in solids. The QSL is perhaps one of the most exotic quantum phases known so far partly because of the nontrivial elementary excitations and has been attracting the attention of condensed matter scientists for several decades. In 2006 a theoretical breakthrough in the field of QSLs was reported. Alexei Kitaev proposed a simple new model that is exactly solvable and that gives a QSL ground state, in which the spins fractionalize into emergent quasiparticles — Majorana fermions　[1]. Soon after, a spin-orbital Jeff = 1/2 Mott insulator was identified in a complex 5d iridium oxide. This led to a theoretical proposal for the realization of the Kitaev model using Jeff = 1/2 pseudo-spins in an iridate, and initiated a search for the QSL state and the hidden Majorana fermions in a family of iridium and ruthenium compounds [2]. I am going to talk about the rapid progress in the materialization Kitaev QSL as well as the hunting of Majorana fermions through the detection of unusual transport. 1. A. Kitaev, Ann. Phys. 321, 2-111 (2006). 2. H.Takagi, T. Takayama, G. Jackeli, G. Khaliullin, S. E. Nagler, Nat. Rev. Phys. 1, 264–280 (2019).

Bilayers of transition metal dichalcogenides (TMDs) have recently emerged as promising platforms to study strongly correlated electrons in two dimensions. In particular, the low-energy physics of these systems can be described by generalized Hubbard models on the triangular lattice. At certain fillings and parameter regimes, generalized Wigner crystals can occur in which the translational invariance of the charge distribution is spontaneously broken. In this work, we investigate TMD heterobilayers at a filling of 3/4 holes per moiré unit cell where the holes are almost entirely localized on a kagome lattice. By expanding the tight-binding model describing the system, we derive an effective spin model on this kagome lattice that includes up to third-neighbor Heisenberg and Dzyaloshinskii–Moriya interactions. Through a combination of density matrix renormalization group simulations and Schwinger boson mean field analysis, we explore the possibilities of realizing quantum spin liquids in this model in experimentally realistic parameter ranges.

The edge of fractional quantum Hall (FQH) states is an outstanding platform for access to strongly correlated electrons with topological properties. Transport of charge and heat on the edge is known to be very robust due to the strong suppression of back-scattering, and has been used to probe fundamental quantum phenomena, e.g., charge fractionalisation and anyonic statistics. However, for edges with counter-propagating modes (e.g. at fillings $\nu=2/3$ or $3/5$) back-scattering is possible and is predicted to induce rich transport characteristics: Depending on the temperature, the transport can range from ballistic to diffusive, and even to full charge-heat separation, where heat and charge propagate ballistically in opposite directions. Understanding such complex behaviour is crucial for determining FQH edge structures and their topological order. In this talk, I describe recent results for probing FQH edges with noise and thermal conductance measurements. First, I describe why and how the detection of excess charge noise on a current biased edge segment is a useful tool for detecting the presence and transport characteristics of counter-propagating edge modes [1]. Second, I present the results of recent experiments [2,3] providing evidence for such features. These results open the door to studies of complex edge transport in more advanced geometries and could also be used to pin-point the topological order of non-Abelian states, e.g. at $\nu=5/2$ [4]. [1] Topological Classification of Shot Noise on Fractional Quantum Hall Edges, C. Spånslätt, J. Park, Y. Gefen, and A.D. Mirlin, Phys.Rev.Lett. 123, 137701 (2019) [2] Observation of ballistic upstream modes at fractional quantum Hall edges of graphene R. Kumar, S.K. Srivastav, C. Spånslätt, K. Watanabe, T. Taniguchi, Y. Gefen, A.D. Mirlin, and A. Das Nat Commun 13, 213 (2022) [3] Experimental control over thermal conductance quantization in fractional quantum Hall states, S.K. Srivastav, R. Kumar, C. Spånslätt, K. Watanabe, T. Taniguchi, A.D. Mirlin, Y. Gefen, and A. Das, arXiv: 2202.00490 [4] Noise on the non-Abelian ν=5/2 fractional quantum Hall edge J. Park, C. Spånslätt, Y. Gefen, and A.D. Mirlin, Phys.Rev. Lett. 125, 157702 (2020)

Flat-band superconductivity demonstrates the importance of band topology and geometry to correlated phases. The geometry of fragile topological and obstructed atomic bands enhances the superfluid weight and hence the superconducting critical temperature. Here, we derive general lower bounds for the superfluid weight in terms of momentum space irreps in all 2D space groups, extending the reach of topological quantum chemistry to superconducting states. Using exact Monte Carlo simulations we prove the validity of our bounds beyond mean-field.

Interactions within topological systems have produced novel phases of matter through the fractional quantum Hall effect and topological Mott phases. Here, we introduce interaction effects between two Bloch spheres each describing a topological model such as a Haldane model or a topological p-wave superconductor. We identify the possibility of fractional topological numbers through the introduction of adjustable Semenoff masses realizing a Z2 sphere-inversion symmetry. We discuss applications and engineering of this classification for topological insulators and topological superconducting wires coupled through a Coulomb interaction. A fractional topological number can also give rise to a topological bilayer semimetal with a quantized π Berry phase at one Dirac point and a nodal ring semimetal at the other Dirac point. The spheres find practical and real applications in circuit quantum electrodynamics and atomic physics through a quantum dynamo effect when rolling from north to south pole.

Superconductor-semiconductor hybrid systems offer an appealing route to topological superconductivity, but many of their basic properties such as density, mobility, and induced pairing are not known. The key challenge is that the proximitizing superconductor acts as a perfect shunt for most electrical probes. To overcome this challenge, we embed a two-dimensional Al-InAs hybrid system in a resonant microwave circuit, probing the breakdown of superconductivity due to an applied magnetic field. We find a fingerprint from the two-component nature of the hybrid system, and quantitatively compare with a theory that includes the contribution of intraband $p \pm i p$ pairing in the InAs, as well as the emergence of Bogoliubov-Fermi surfaces. Separately resolving the Al and InAs contributions additionally allows us to determine the carrier density and mobility in the InAs.

The simultaneous interplay of strong electron-electron correlations, topological zero-energy states, and disorder is still a largely unexplored territory but of considerable interest due to their inevitable presence in many materials. Copper oxide high-temperature superconductors (cuprates) with pair breaking edges host a flat band of topological zero-energy states, making them an ideal playground where strong correlations, topology, and disorder are strongly intertwined. Here we perform fully self-consistent calculations of the superconducting state in cuprate superconductors and find that strong correlations stabilize a fully gapped phase crystal state along the pair breaking [110] edge. The phase crystal breaks both translational and time reversal invariance and is characterized by a nanoscale modulation of the phase of the d-wave order parameter. In particular, we first show how strong correlations increase the number of zero-energy states for any uniform superconducting phase, contradicting simple topological arguments. Then, when allowing for a non-uniform solution, we find a cascade of phase transitions occurring at different temperatures: d-wave superconductivity occurs below a transition temperature $T_c$, the phase crystal appears further below at temperature $T^* \sim 0.2 T_c$, and, finally, an additional extended s-wave order, with the same spatial modulations as the phase crystal, is generated below $T_s$, producing a full energy gap. Taken together, these phase transitions explain a set of so far seemingly contradictory experimental results. Furthermore, we find that the phase crystal state is unexpectedly very robust to disorder, but notably only in the presence of strong correlations. Our results show that the combined effects of strong correlations and topology lead to the emergence of novel phases of matter that survives strong disorder in a highly non-intuitive manner.

The Jordan-Wigner transformation provides a manifestly local representation of a lattice fermionic system using bosonic degrees of freedom. The bosonic description could provide an alternative starting point for tackling problems concerning correlated electrons in both numerical studies and, possibly, near-term intermediate-scale quantum simulators. However, a direct application of the original Jordan-Wigner transformation to higher than one spatial dimension suffers from non-locality. Local generalizations of the transformation to higher dimensions exist, but they either lead to unnatural symmetry representations, or require an excessive number of auxiliary degrees of freedom. In this talk, we discuss how one could perform Jordan-Wigner transformation in higher dimensions which respect keep the locality and symmetries manifest.

We supplement the notions of stable and fragile topology by introducing delicate topological insulators: band structures possessing topological invariants that can be trivialized through an addition of a trivial band to either valence or conduction subspace. In my talk I will present a family of delicate topological insulators protected by rotational symmetry. I will discuss their unusual topological classification given by complementary topological invariants, which are at the same time related to each other modulo the protecting rotation symmetry. I will also introduce bulk-boundary correspondence of the studied delicate models, which differs from bulk-boundary correspondence known for stable topological insulators and manifests in anomalous symmetry eigenvalues of the surface states. These anomalous eigenvalues lead to conducting states at sharply terminated boundaries of delicate topological insulators.

Non-trivial topological phases are much pursued by the researchers in condensed matter physics due to its novel transport properties, which may enable the development of future technologies in the field of spintronics. Furthermore, the addition of Ce ions in the compounds leads to other complex properties, such as crystalline electrical field effects, magnetism, and the Kondo effect. The interaction of these effects with topological phases are not fully understood yet and might enhance the observation of unusual transport properties, for instance, the Kondo effect may pin band crossings close to the Fermi energy favoring non-trivial topological properties [1]. In particular, the family of compounds $Ln$Al$X$ ($Ln =$ lanthanides, $X =$ Ge, Si) is promising to host non-trivial topological phases. Its members crystallize in the noncentrosymmetric structure ($I4_{1}md$), which breaks the space-inversion symmetry, and many present magnetic orders, that break time-reversal symmetry. The breaking of these symmetries are the main ingredients to the formation of magnetic Weyl semimetals. Here we focus on CeAlSi, in which a noncollinear ferromagnetic order takes place below 8.2~K [2] and chiral domain walls were recently observed [3,4]. We investigated the non-trivial topological properties of CeAlSi by combining electrical resistivity, Hall effect, and magnetization measurements with pressure tuning and DFT calculations [5]. A large contribution of domain wall scattering to the anomalous Hall effect and an atypical temperature response of the quantum oscillations amplitude were observed. Applying external pressure suppresses both behaviors; whereas it enhances $T_{C}$ to 9.4~K at 2.7~GPa. Magnetization measurements show no evidence of changes in the magnetic structure as a function of pressure. Finally, DFT calculations revealed a negligible effect of external pressure on the position of the Weyl nodes. Therefore, we suppose that the suppression of the anomalous responses of the Hall effect and the quantum oscillations amplitude are related to changes in the domain wall landscape of CeAlSi. [1] S. E. Grefe, H.-H. Lai, S. Paschen, and Q. Si, Phys. Rev. B, 101, 075138 (2020). [2] H.-Y. Yang et al., Phys. Rev. B, 103, 115143 (2021). [3] B. Xu, J. Franklin, A. Jayakody, H.-Y. Yang, F. Tafti, I. Sochnikov, Adv. Quantum Technol., 4, 2000101 (2021). [4] Y. Sun, C. Lee, H.-Y. Yang, D. H. Torchinsky, F. Tafti, J. Orenstein, Phys. Rev. B, 104, 235119 (2021). [5] M. M. Piva, J. C. Souza, V. Brousseau-Couture, K. R. Pakuszewski, Janas K. John, C. Adriano, M. Côté, P. G. Pagliuso, M. Nicklas, arXiv:2111.05742v1 (2021).

Physics is an empirical science. Thus, experiments play a fundamental role in what we know and understand about reality. In this brief talk, I introduce the concept of chiral multifold fermions: low-energy quasiparticles emerging near multifold band crossings protected by the crystal symmetries of the material. I propose optical probes to detect chiral multifold fermions in real materials and provide explicit examples with the multifold semimetals RhSi and CoSi. I conclude by pointing out the current challenges in using optical probes to identify multifold fermions, both theoretically and experimentally.

Various phenomena occur when two-dimensional materials, such as graphene or transition metal dichalcogenides, are assembled into bilayers with a twist between the individual layers. As an application of this paradigm, we predict that structures composed of two-monolayer-thin $d$-wave superconductors with a twist angle can form a gapped topological phase with spontaneously broken time-reversal symmetry and protected chiral Majorana edge modes. These structures can be realized by mechanically exfoliating van der Waals-bonded high-$T_c$ cuprates, such as BSCCO. Symmetry arguments and microscopic modelling suggest that this phase will form for a range of twist angles in the vicinity of 45$^{\circ}$, and will set in at a temperature close to the bulk superconducting critical temperature of 90K. Therefore, the platform may provide a realization of a high-temperature topological superconductor. Ref: O Can, T Tummuru, R Day, I Elfimov, A Damascelli and M Franz, ‘High-temperature topological superconductivity in twisted double-layer copper oxides’ Nat Phys 17, 519 (2021).

Magic angle twisted bilayer graphene (MATBG) is a 2-dimensional material which has emerged as a new exciting platform to study strongly correlated physics. It presents a simultaneous co-existence and gate-tunability of different phases including superconductivity, magnetism and non-trivial topological orders. This opens up entirely new possibilities for the creation of complex hybrid Josephson junctions (JJ). Here we report on the creation of gate-defined, magnetic Josephson junctions in MATBG, where the weak link is gate-tuned close to the correlated state at a moiré filling factor of ν = −2. A highly unconventional Fraunhofer pattern emerges, which is phase-shifted and asymmetric with respect to the current and magnetic field directions, and shows a pronounced magnetic hysteresis. We demonstrate how the combination of magnetization and current induced magnetization switching in the MATBG JJ allows us to realize a programmable zero field superconducting diode.

The search for Majorana quasiparticles is one of the leading topics in the current solid-state physics research. The promising device to be a host for Majorana zero modes is a one-dimensional topological superconducting nanowire, the idea proposed by Kitaev at the turn of the century. The signatures of the Majorana bound states may be identified within the spectral and transport properties of hybrid zero- and one-dimensional devices. Such a device can be a combination of a Majorana wire and a single or double quantum dot. In this communication, I want to shed some light on the electronic and thermoelectric transport properties of a double quantum dot coupled with a one-dimensional topological superconducting nanowire and show, how these properties may be modified in the presence of such exotic quasiparticles. All of the results are obtained in the strongly correlated regime when the Kondo effect occurs. The work is based on the numerical renormalization group procedure, a reliable theoretical method for solving quantum impurity problems, where the interactions are present.

Recent theoretical predictions and experimental realizations of exotic fermions and topologically protected phases in condensed matter have led to tremendous research interests in topological quantum materials. Especially, correlated topological materials, such as magnetic Dirac and Weyl semimetals, and intrinsic magnetic topological insulators etc., in which both non-trivial topology of single-electron band structures and electronic correlation effects are essential for the underlying physics, have emerged as an exciting platform to explore novel electronic and magnetic phenomena. In this talk, I will present a couple of the selected examples of our recent neutron scattering studies of correlated topological materials, including magnetic Dirac semimetal EuMnBi$_2$ [1], magnetic Weyl semimetal Mn$_3$Sn and topological magnon insulators in two-dimensional van der Waals ferromagnets CrSiTe$_3$ and CrGeTe$_3$ [2]. [1] Fengfeng Zhu, et al., Phys. Rev. Research 2, 043100 (2020). [2] Fengfeng Zhu, et al., Sci. Adv. 7, eabi7532 (2021).

In this talk I present a toy model, developed together with my collaborators, of the surface electronic band structure of the material NbGeSb. This band structure shows an unusual '3+1' motif along the edge of the surface Brillouin zone, where a Rashba-split pair of holelike bands crosses a Rashba-split pair of electron-like ones but only one of the four crossing points appears to develop a gap. The toy model explains this feature in terms of an interesting interplay between the spin and orbital content of the bands. A result of the model is the existence of Weyl-like points where it is the orbital angular momentum rather than the spin angular momentum that exhibits a non-trivial winding as the crossing point is encircled.

Half-quantum vortices (HQVs) have a Majorana bound state associated with them that mean they obey non-Abelian statistics. When moved around each other this generates a quantum entanglement that could be exploited to make a robust quantum computer. Stable half quantum vortices (HQVs) have been observed in geometrically confined superfluid helium-3 and in polariton condensates. The observation of mobile HQVs in superconductors however remains elusive. Localised half-flux quanta have been detected in unconventional superconductors at tri-crystal grain boundaries in the cuprates and arguably in mesoscopic rings of Sr$_2$RuO$_4$and polycrystalline $\beta$-Bi$_2$Pd. However, the non-integer flux in these cases is a consequence of the samples engineered geometry. In this poster/talk we will explore which materials could be favourable candidates to host unconfined half quantum vortices.

Topology, a mathematical concept, recently became a hot and truly transdisciplinary topic in condensed matter physics, solid state chemistry and materials science. All 200 000 inorganic materials were recently classified into trivial and topological materials: topological insulators, Dirac, Weyl and nodal-line semimetals, and topological metals [1]. Around 20% of all materials host topological bands. Currently, we have focussed also on magnetic materials, a fertile field for new since all crossings in the band structure of ferromagnets are Weyl nodes or nodal lines [2], as for example Co2MnGa and Co3Sn2S2. Beyond a single particle picture and identified antiferromagnetic topological materials [3]. 1. Bradlyn et al., Nature 547 298, (2017), Vergniory, et al., Nature 566 480 (2019). 2. Belopolski, et al., Science 365, 1278 (2019), Liu, et al. Nature Physics 14, 1125 (2018), Guin, et al. Advanced Materials 31 (2019) 1806622, Liu, et al., Science 365, 1282 (2019), Morali, et al., Science 365, 1286 (2019) 3. Xu et al. Nature 586 (2020) 702.

A key ingredient to obtaining topological quantum devices is the ability to control topological states without modifying the underlying properties of the material that hosts them. It is well known that in heavy fermion metals physical properties can be much more readily tuned than in simple metals. Such correlated materials may also exhibit extreme signatures of nontrivial topology, as recently demonstrated for the Weyl-Kondo semimetal Ce$_3$Bi$_4$Pd$_3$ [1, 2, 3]. What has remained open, however, is whether these topological states are essentially inert, or can also be influenced by external control parameters. Here we show the latter to be the case: the material's Weyl signatures can be fully suppressed by a relatively modest magnetic field [4]. We understand this behavior as a Zeeman-driven motion of Weyl nodes in momentum space, up to the point where the nodes meet and annihilate in a topological quantum phase transition. The topologically trivial but correlated background remains largely unaffected across this transition. Only at larger fields a transition from a correlated narrow-gap phase to a metallic heavy Fermi liquid phase is observed. Our work lies at the scarcely explored intersection between gapless topology and strong correlations, and accentuates the importance of materials innovation to obtain much needed breakthroughs in topological quantum applications. [1] S.\,Dzsaber \textit{et al.}, \textit{Phys.\,Rev.\,Lett.\,}{\bf 118}, 246601 (2017).\\ [2] H.\,-H.\,Lai \textit{et al.}, \textit{Proc.\,Natl.\,Acad.\,Sci.\,U.S.A.\,}{\bf 115}, 93 (2018).\\ [3] S.\,Dzsaber \textit{et al.}, \textit{Proc.\,Natl.\,Acad.\,Sci.\,U.S.A.\,}{\bf 118}, e2013386118 (2021).\\ [4] S.\,Dzsaber \textit{et al.}, \textit{arXiv:1906.01182} (2019).\\

Over the last few years, gapless topological phases driven by strong correlations have gained considerable prominence. In particular, the theoretical concept and key properties of Weyl-Kondo semimetal have been advanced [1,2]. These ideas set the stage for the development of a general framework that allows for identifying novel phases and realizing them in new materials. In this talk, I will present a general approach [3,4] in which strong correlations cooperate with crystalline symmetry to drive gapless topological states. I will validate this approach by exploring Kondo lattice models and materials whose space group symmetries may promote different kinds of electronic degeneracies, with a particular focus on square-net systems. In the process, I will identify Weyl-Kondo Nodal-Line semimetals where nodal-line excitations are driven by the Kondo effect [3]. I will further demonstrate how strong correlations cooperate with lattice symmetry to produce gapless topological phases that have no Landau quasiparticles and show strange-metal behavior [4]. Finally, I will outline a general procedure to identify materials for the realization of these correlation-driven topological semimetal phases. The findings presented in this talk will illustrate the potential of the proposed materials design principle to guide the search for new topological phases and materials in a broad range of strongly correlated systems. [1] H.-H. Lai et al, PNAS115, 93 (2018); S. E. Grefe et al., Phys. Rev. B101, 075138 (2020). [2] S. Dzsaber et al., Phys. Rev. Lett. 118, 246601 (2017); PNAS 118,e2013386118(2021). [3] L. Chen et al, arXiv:2107.10837. [4] H. Hu et al., arXiv:2110.06182.

The notion of an electronic flat band refers to a collectively degenerate set of quantum mechanical eigenstates in periodic solids. The vanishing kinetic energy of flat bands relative to the electron-electron interaction is expected to result in a variety of many-body quantum phases of matter. Here we present recent developments in realizing flat bands in transition element-based kagome metals. We will present recent experiments in which a partial filling of a flat band is associated with unusual transport and thermodynamic that recall those of strongly correlated systems. We will also comment on the potential role of band topology and prospects for using similar lattice and orbital engineering to realize new correlated systems.

In this talk, the concept and properties of topological Kondo insulators are reviewed [1]. The focus will be on SmB$_6$, which possesses topologically protected nontrivial surface states inside the bulk hybridization gap. Experimentally, hybridization between localized 4$f$ and conduction band states at sufficiently low temperatures is well established [2]. Scanning tunneling microscopy and spectroscopy (STM/STS) were able to identify the surface states within the hybridization gap (especially below 5 K) and a surface Kondo breakdown as well as the local suppression of the surface states by impurities of magnetic and non-magnetic nature [3,4]. Results from ARPES and quantum oscillation measurements are also discussed [5]. Moreover, the surface states give rise to interesting transport properties. Other materials proposed as topological Kondo insulators will also be reviewed briefly. Work conducted in collaboration with M. V. Ale Crivillero, Z. Fisk, L. Jiao, P. Schlottmann, S. Rößler, P. F. S. Rosa, F. Steglich and L. H. Tjeng. [1] S. Wirth and P. Schlottmann, Adv. Quantum Technol. {\bf 4}, 2100102 (2021). [2] S. Rößler et al., Proc. Natl. Acad. Sci. USA {\bf 111}, 4798 (2014). [3] Lin Jiao et al., Nature Commun. {\bf 7}, 13762 (2016). [4] Lin Jiao et al., Sci. Adv. {\bf 4}, eaau4886 (2018). [5] Lu Li et al., Nat. Rev. Phys. {\bf 2}, 463 (2020).

Quantum oscillations are an important tool to understand the electronic behaviour of novel quantum materials. Besides giving access to the size of the Fermi surfaces and their shapes, quantum oscillations provide information about the quasiparticle effective masses, quantum scattering rates and spin-splitting effects. Additionally, the phase accumulated in quantum oscillations contains information on the origin of the orbits including information about Berry phases. In this talk I will provide a broad introduction to quantum oscillations and discuss the experimental data for different topological materials. I will also present the effects of the Fermi surface reconstruction in the presence of a magnetic order that can induce topological effects.